Modules for loop Affine-Virasoro algebras

Highest Weight Module ◽

Highest Weight ◽

Finite Dimensional ◽

Integrable Module ◽

In this paper, we study the representations of loop Affine-Virasoro algebras. As they have canonical triangular decomposition, we define Verma modules and their irreducible quotients. We give necessary and sufficient condition for a irreducible highest weight module to have finite dimensional weight spaces. We prove that an irreducible integrable module is either a highest weight module or a lowest weight module whenever the canonical central element acts non-trivially. At the end, we construct Affine central operators for each integer and they commute with the action of the Affine Lie algebra.

Quantum Groups: Singular Vectors and BGG Resolution

International Journal of Modern Physics A ◽

10.1142/s0217751x92003963 ◽

1992 ◽

Vol 07 (supp01b) ◽

pp. 623-643 ◽

Cited By ~ 5

Author(s):

Fyodor Malikov

Keyword(s):

Quantum Group ◽

Quantum Groups ◽

Weight Module ◽

Verma Modules ◽

Highest Weight Module ◽

Singular Vectors ◽

Type Formula ◽

Highest Weight ◽

Finite Dimensional ◽

Dimensional Module

We prove existence of BGG resolution of an irreducible highest weight module over a quantum group, classify morphisms of Verma modules over a quantum group and find formulas for singular vectors in Verma modules. As an application we find cohomology of the quantum group of the type [Formula: see text] with coefficients in a finite-dimensional module.

A Family of Non-weight Modules over the Virasoro Algebra

Algebra Colloquium ◽

10.1142/s100538672000067x ◽

2020 ◽

Vol 27 (04) ◽

pp. 807-820

Author(s):

Guobo Chen

Keyword(s):

Tensor Product ◽

Sufficient Conditions ◽

Virasoro Algebra ◽

Necessary And Sufficient Conditions ◽

Weight Module ◽

Highest Weight Module ◽

Isomorphism Classes ◽

Highest Weight ◽

Necessary And Sufficient ◽

Weight Modules

In this paper, we consider the tensor product modules of a class of non-weight modules and highest weight modules over the Virasoro algebra. We determine the necessary and sufficient conditions for such modules to be simple and the isomorphism classes among all these modules. Finally, we prove that these simple non-weight modules are new if the highest weight module over the Virasoro algebra is non-trivial.

GENERIC IRREDUCIBLE REPRESENTATIONS OF FINITE-DIMENSIONAL LIE SUPERALGEBRAS

International Journal of Mathematics ◽

10.1142/s0129167x9400022x ◽

1994 ◽

Vol 05 (03) ◽

pp. 389-419 ◽

Cited By ~ 33

Author(s):

IVAN PENKOV ◽

VERA SERGANOVA

Keyword(s):

Irreducible Module ◽

Lie Superalgebra ◽

Lie Superalgebras ◽

Irreducible Representations ◽

Sufficient Condition ◽

Highest Weight ◽

Finite Dimensional ◽

Finite Dimensionality ◽

A theory of highest weight modules over an arbitrary finite-dimensional Lie superalgebra is constructed. A necessary and sufficient condition for the finite-dimensionality of such modules is proved. Generic finite-dimensional irreducible representations are defined and an explicit character formula for such representations is written down. It is conjectured that this formula applies to any generic finite-dimensional irreducible module over any finite-dimensional Lie superalgebra. The conjecture is proved for several classes of Lie superalgebras, in particular for all solvable ones, for all simple ones, and for certain semi-simple ones.

Quasi-finite Irreducible Graded Modules for the Virasoro-like Algebra

Algebra Colloquium ◽

10.1142/s1005386713000175 ◽

2013 ◽

Vol 20 (02) ◽

pp. 181-196 ◽

Cited By ~ 4

Author(s):

Weiqiang Lin ◽

Yucai Su

Keyword(s):

Weight Module ◽

Highest Weight Module ◽

Graded Modules ◽

Highest Weight ◽

Finite Dimensional ◽

Irreducible Modules ◽

Graded Module ◽

Intermediate Series ◽

Uniformly Bounded

In this paper, we consider the classification of irreducible Z- and Z2-graded modules with finite-dimensional homogeneous subspaces over the Virasoro-like algebra. We prove that such a module is a uniformly bounded module or a generalized highest weight module. Then we determine all generalized highest weight quasi-finite irreducible modules. As a consequence, we determine all the modules with nonzero center. Finally, we prove that there does not exist any non-trivial Z-graded module of intermediate series.

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

On the forces that cable webs under tension can support and how to design cable webs to channel stresses

10.1098/rspa.2018.0781 ◽

2019 ◽

Vol 475 (2223) ◽

pp. 20180781 ◽

Cited By ~ 5

Author(s):

Guy Bouchitté ◽

Ornella Mattei ◽

Graeme W. Milton ◽

Pierre Seppecher

Keyword(s):

Linear Programming ◽

Convex Hull ◽

Structural Engineering ◽

Three Dimensional ◽

Two Dimensions ◽

Three Dimensions ◽

Sufficient Condition ◽

Finite Dimensional ◽

In many applications of structural engineering, the following question arises: given a set of forces f 1 , f 2 , …, f N applied at prescribed points x 1 , x 2 , …, x N , under what constraints on the forces does there exist a truss structure (or wire web) with all elements under tension that supports these forces? Here we provide answer to such a question for any configuration of the terminal points x 1 , x 2 , …, x N in the two- and three-dimensional cases. Specifically, the existence of a web is guaranteed by a necessary and sufficient condition on the loading which corresponds to a finite dimensional linear programming problem. In two dimensions, we show that any such web can be replaced by one in which there are at most P elementary loops, where elementary means that the loop cannot be subdivided into subloops, and where P is the number of forces f 1 , f 2 , …, f N applied at points strictly within the convex hull of x 1 , x 2 , …, x N . In three dimensions, we show that, by slightly perturbing f 1 , f 2 , …, f N , there exists a uniloadable web supporting this loading. Uniloadable means it supports this loading and all positive multiples of it, but not any other loading. Uniloadable webs provide a mechanism for channelling stress in desired ways.

A necessary and sufficient condition for irreducibility of parabolic baby Verma modules of sl(4,k)

Journal of Pure and Applied Algebra ◽

10.1016/j.jpaa.2014.04.028 ◽

2015 ◽

Vol 219 (4) ◽

pp. 760-766 ◽

Cited By ~ 2

Author(s):

Yi-Yang Li ◽

Bin Shu ◽

Yu-Feng Yao

Keyword(s):

Sufficient Condition ◽

Verma Modules ◽

On the Vanishing of a (G, σ) Product in a (G, σ) Space

Canadian Journal of Mathematics ◽

10.4153/cjm-1970-044-4 ◽

1970 ◽

Vol 22 (2) ◽

pp. 363-371 ◽

Cited By ~ 6

Author(s):

K. Singh

Keyword(s):

Vector Space ◽

Multiplicative Group ◽

Cartesian Product ◽

Arbitrary Field ◽

Tensor Space ◽

Linear Character ◽

Finite Dimensional Vector Space ◽

Finite Dimensional ◽

In this paper, we shall construct a vector space, called the (G, σ) space, which generalizes the tensor space, the Grassman space, and the symmetric space. Then we shall determine a necessary and sufficient condition that the (G, σ) product of the vectors x1, x2, …, xn is zero.1. Let G be a permutation group on I = {1, 2, …, n} and F, an arbitrary field. Let σ be a linear character of G, i.e., σ is a homomorphism of G into the multiplicative group F* of F.For each i ∈ I, let Vi be a finite-dimensional vector space over F. Consider the Cartesian product W = V1 × V2 × … × Vn.1.1. Definition. W is called a G-set if and only if Vi = Vg(i) for all i ∊ I, and for all g ∊ G.

Cylindrical representations of some infinite dimensional nuclear Lie groups

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700011916 ◽

1992 ◽

Vol 46 (2) ◽

pp. 295-310 ◽

Cited By ~ 1

Author(s):

Jean Marion

Keyword(s):

Lie Groups ◽

Lie Group ◽

Additive Group ◽

Test Functions ◽

Infinite Dimensional ◽

Finite Dimensional ◽

Irreducible Unitary Representations ◽

Direct Product Group ◽

Let Γ.𝒜 be the semi-direct product group of a nuclear Lie group Γ with the additive group 𝒜 of a real nuclear vector space. We give an explicit description of all the continuous representations of Γ.𝒜 the restriction of which to 𝒜 is a cyclic unitary representation, and a necessary and sufficient condition for the unitarity of such cylindrical representations is stated. This general result is successfully used to obtain irreducible unitary representations of the nuclear Lie groups of Riemannian motions, and, in the setting of the theory of multiplicative distributions initiated by I.M. Gelfand, it is proved that for any connected real finite dimensional Lie groupGand for any strictly positive integerkthere exist non located and non trivially decomposable representations of orderkof the nuclear Lie group(M;G) of all theG-valued test-functions on a given paracompact manifoldM.